A058364 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 9 sites wide.

1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 28, 39, 51, 64, 78, 93, 109, 126, 144, 172, 211, 262, 326, 404, 497, 606, 732, 876, 1048, 1259, 1521, 1847, 2251, 2748, 3354, 4086, 4962, 6010, 7269, 8790, 10637, 12888, 15636, 18990, 23076, 28038

Offset

1,9

Comments

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

References

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.

Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Links

Table of n, a(n) for n=1..53.

Formula

a(n) = 1 + n*sum(binomial(n-1-8*i, i-1)/i, i=1..n/9). a(n) = a(n-1) + a(n-9), a(n) = 1 for n = 1..8, a(9) = 10. generating function = (x+9*x^9)/(1-x-x^9).

Example

a(9) = 10 because there is one way to put zero molecule to the necklace and 9 ways to put one molecule.

Crossrefs

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

Sequence in context: A008715 A115844 A008714 * A228915 A162790 A008713

Adjacent sequences: A058361 A058362 A058363 * A058365 A058366 A058367

Keyword

nonn,changed

Author

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

Status

approved